My mother-in-law is a wonderful quilter and when her father celebrated his 95th birthday she made him a quilt with pictures of him when he was younger. I can't remember what the stuff was called, but I do know you can find it at any Hancock fabric, or Joann's or Michael's store.
What might work for you is to have the kids draw pictures on paper and then you scan those pictures into your computer and print them out. I believe it actually prints onto the fabric, but I can't be sure. I can't see my MIL going through all those steps of scanning picture in, printing onto iron on, transfering, and then piecing together, so I'm pretty sure it went right onto some type of muslin (or white anyway) fabric.
Looking at this, I don't seem to be too helpful, but back then I was still new to the family so I didn't know about it until the party. But I do know that if you go to any fabric or craft store they will help you.
I know for my kids' godparents Christmas gifts I traced their hands with Crayola fabric markers onto a flour sack dish towel. I'm pretty sure none of the godparents have actually used the dish towel, but it's another idea.
Also, how big of a quilt are you looking at making, just a small wall hanging type or baby quilt or larger? If it's not something that will actually be washed very often I think you can get away with using the fabric markers.
I hope I've been some sort of help!! It sounds like such a cute idea! Good luck!